Generalized energy and gradient flow via graph framelets
This work addresses the challenge of enhancing representation power in graph neural networks for tasks involving both homophilic and heterophilic graphs, though it appears incremental as it builds on existing framelet-based models.
The authors tackled the problem of understanding and improving graph neural networks (GNNs) by theoretically analyzing framelet-based models through energy gradient flow, showing they induce both low- and high-frequency dynamics, which explains good performance on homophilic and heterophilic graphs, and proposed a generalized energy leading to a novel GNN that includes existing models as special cases.
In this work, we provide a theoretical understanding of the framelet-based graph neural networks through the perspective of energy gradient flow. By viewing the framelet-based models as discretized gradient flows of some energy, we show it can induce both low-frequency and high-frequency-dominated dynamics, via the separate weight matrices for different frequency components. This substantiates its good empirical performance on both homophilic and heterophilic graphs. We then propose a generalized energy via framelet decomposition and show its gradient flow leads to a novel graph neural network, which includes many existing models as special cases. We then explain how the proposed model generally leads to more flexible dynamics, thus potentially enhancing the representation power of graph neural networks.